Jacobi Triple Product

In mathematics, the Jacobi triple product is the mathematical identity:

\prod_{m=1}^\infty \left( 1 - x^{2m}\right) \left( 1 + x^{2m-1} y^2\right) \left( 1 + x^{2m-1} y^{-2}\right) = \sum_{n=-\infty}^\infty x^{n^2} y^{2n}.

for complex numbers x and y, with |x| < 1 and y ≠ 0.

It was introduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum.

The Jacobi triple product identity is the Macdonald identity for the affine root system of type A1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra.

The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity.

Let and . Then we have

\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) = \sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.\,

The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:

Let and .

Then the Jacobi theta function

\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z).

can be written in the form

Using the Jacobi Triple Product Identity we can then write the theta function as the product

\vartheta(z; \tau) = \prod_{m=1}^\infty \left( 1 - \exp(2m \pi i \tau)\right) \left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right) \left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right).

There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of q-Pochhammer symbols:

\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n = (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.

Where is the infinite q-Pochhammer symbol.

It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For it can be written as

Read more about Jacobi Triple Product:  Proof

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